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Strong limit theorems of empirical distributions for large segmental exceedances of partial sums of Markov variables. (English) Zbl 0746.60029

In the paper reviewed above the authors characterized the composition of high scoring segments among partial sums of i.i.d. random variables. In this paper the authors consider a letter sequence A 1 ,,A n assuming values from a finite alphabet {a i } i=1,,r governed by an r-state irreducible Markov chain. Suppose (X m ,U m ) are independently distributed given the sequence A 1 ,A 2 ,, where the joint distribution of (X m ,U m ) depends only on A m-1 and A m and is of bounded support. When A 0 is started with its stationary distribution, EX 1 <0 and the existence of a finite cycle C={A 0 =i 0 ,,A k -i k =i 0 } such that PS m = i=1 m X i > 0 , m = 1 , , k ; C>0 is assumed. Define the stopping times

K 0 =0,K ν =mink K ν-1 + 1 , S k - S K ν-1 0,ν1,

and for y>0,

T ν (g)=minm : m > K ν-1 and either S m - S K ν-1 0 or S m - S K ν-1 y,
L ν (y)=T ν (g)-K ν-1 ,W ν (g)= m=K ν-1 +1 T ν (g) U m ·

Theorems 1 and 2: Let I ν (y)=1 while I 1 (y)==I ν-1 (y)=0. Then L ν (y)/y1/w * a.s. as t, and W ν (y)/L ν (g)u * a.s. as y, for any value of A 0 , certain positive constants w * , u * .

60F15Strong limit theorems
60F10Large deviations
60K15Markov renewal processes
60J10Markov chains (discrete-time Markov processes on discrete state spaces)